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Full text of "Mathematical And Physical Papers - Iii"

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H'Y%, or, on account of the smallness of cr, in the ratio of 1 to 1 + i (!'!-* +H'ff*) nearly. Now %H'H~l is the correction for buoyancy, and therefore
We have also, if ki be the value of the function k of Section III., Part L, P = kaV(l -f a)2 4- ^k^Yf,     H' = crF(Z + a) + -JcrF^ ... (150),
and jBT"1 = (Z 4- a)'1 very nearly. Substituting in (149), expanding the denominator, and neglecting Ft2, we get
Vl    /   Z   \2    1 Vl,    Z
Now Ft is very small compared with F, and it is only by being multiplied by the large factor k^ that it becomes important. We may then, without any material error, replace the last term in the above equation by J FjF^Z2 (Z -f a)~'\ and if X be the length of the isochronous simple pendulum, we may suppose Z -f a = X, and replace Z2 (Z + a)"2 by 1  2aX~1, since a is small compared with X. We thus get, putting An for the correction due to the wire,
Substituting for &A   1 from (115), and for ttt from (147), in which equations, however, kl9 c^ must be supposed to be written for ky a, expressing FI} Fin terms of the diameters of the wire and sphere, and neglecting as before a2 in comparison with X5*, we get

It is by these formulae that I have computed the correction for the wire in the following table. In the experiments, the time of oscillation was so nearly one second that it is sufficient in the formulae (148), (151), and (152) to put r= 1, and take X for the length of the seconds' pendulum, or 3914 inches.